Since my previous post, I have received a number of comments from people that suggest they don’t understand what mathematics is. Although there is no single, accepted definition, I think it’s important for maths teachers and educationalists to have a fairly clear idea of what maths is otherwise we risk innovating our way out of teaching it.
In short, my claim is that mathematics uses deductive reasoning whereas areas such as science and history use inductive reasoning. This means that mathematical results are certain in a way that scientific or historical truths are not. Instead, the latter are probabilistic. Here is a useful primer on inductive versus deductive reasoning. Critically, the certainty that mathematics provides has a pretty big flaw – truths are only true if the axioms they are based upon are true and mathematics cannot be used to prove its own starting axioms.
To give an example of deductive reasoning:
God favours the English
This woman is English
Therefore, God favours this woman
The logic of the syllogism is complete and flawless, but this does not mean that God favours the English. Such a question is obviously outside the realms of deductive reasoning (note that I am not claiming that all deductive reasoning is mathematics – I am using these syllogisms because they often make the point more clearly than a mathematical example).
This is similar to the supposedly subjective bits of maths that people point me towards, such as an assumption about a base-rate probability or whether a data point is an outlier. These questions are scientific, in the broadest sense. For instance, computing p-values for psychology experiments has become controversial in recent years. Nevertheless, the deductive logic involved in computing them remains the same. The controversy lies in the inferences we draw on the basis of these values and that, again, is a scientific question. We cannot mathematically prove the validity or otherwise of the p-value.
To give another example, suppose we have a sample of 200 people and we know that 5 have influenza, we can then do plenty of maths. We can determine the standard deviation, work out confidence intervals and so on. The axioms we use come from the data and from assumptions that sit behind standard deviations and confidence intervals. We cannot mathematically prove the value of confidence intervals and we cannot mathematically prove any scientific conclusions. However, these values may be used to inform scientific, inductive reasoning.
Essentially, applied maths makes use of maths in answering other kinds of questions. There is a maths part and a non-maths part to this process. If you are a trained scientist in the heat of problem solving, you may not see or even care about the distinction, but there is one.
Clearly, there are plenty of questions that cannot be addressed by mathematics and there are paradoxes within mathematics where mathematical reasoning breaks down (an example of a paradox is that you cannot prove the truth or otherwise of the statement ‘this statement is false’ using deductive reasoning). However, neither of these introduce matters of opinion as part of mathematical reasoning and neither of these falsify my claim that mathematics is right or wrong. That’s like suggesting the claim that ‘a car is a form of transport’ is false because cars sometimes break down or cars cannot transport minke whales.
Similarly, Gödel’s Theorem does not prove the subjectivity of mathematics, despite what some have claimed. Ironically for those who attempt to make this argument, Gödel’s theorem is itself a beautiful piece of deductive reasoning. It demonstrates that whatever formal system of mathematics we devise, there will be mathematical truths that cannot be demonstrated to be true within that system. This provides a theoretical limit on what we can prove within any given formal system. It does not introduce subjectivity.
Filling the pail 
This strikes me as a “no true Scotsman” argument. You define Mathematics as one thing, then anything that isn’t that isn’t Mathematics. Because you say so.
But I’m a Maths teacher, and what I teach includes what you say isn’t Mathematics. As my curriculum demands, I hasten to add. That you think my non-objective deductions are “science” because they don’t meet your definition of Mathematics is, to put it mildly, a minority opinion.
Interestingly, the definition of maths is a matter of opinion. You are entitled to your view and I respect it, while disagreeing with it. To me, definitions serve to make useful distinctions. I already have the concept of science to cover science and so that’s why it is important to me to preserve what is unique about maths.
I agree with Greg on both counts. This is a debate about word usage so open to differing opinions. And even if you teach some application of math in a math course you are teaching how an abstract set of ideas ( the math part) applies to a real world situation (the non-math part).
Normally the distinction is not important because both are covered and names are not the thing. But whenever someone wants to claim they are teaching math and they are entirely missing the true math part (the abstract axioms and theories or the results of these) it is worth point out they have entirely missed the math portion.
God doesn’t favour the English. Look at their weather!
More seriously: something like ‘numerical based modelling and examination of reality, or what reality might be…
A clever description of mathematics is that is the study of tautologies. That all the theorems of say number theory say the same thing as the axioms because there is no alternative but that the theorems are true if the axioms are.
Now it is fair to say that many would call that pure mathematics and that applied mathematics extends the subject to applications with real world inductively derived ideas and that this is still the study of mathematics not say physics.
But then how do you separate any of these – when is it physics and when is it applied math? If you ask when would you ask a physicist and when would you ask a mathematician the answer would be the physicist if I am interested in the empirical theories and experimental results and the mathematician when I want to be sure the treatment of these is rigorous in the sense that the deductive theories are applied correctly – that is I am not applying some mathematical theory where the assumptions for it don’t hold.
I like your answer Harry.
These are pretty cool epistemological questions around domain boundaries that are primarily studied in philosophy but which can’t be separated from mathematics.
Coupled with a somewhat imprecise duality (objective/subjective) I doubt many mathematicians would want to spend time considering such issues and they would likely aknowledge the primacy of philosophy in approaching such a problem.
I also think Chester nailed it with the no true Scotsman reference. We have had to redefine objective as deductive to keep the initial principle alive. I think treating those two terms as perfectly synonymous is clearly preposterous.
The fact that mathematics uses some of the cleanest examples of logical (deductive reasoning) dosn’t excuse it from the fundamental limitations of all knowledge. In fact I would poetically argue that mathematics (and science) is the study of islands of solidity in a sea of quicksand and that’s all.
If that isn’t enough the word objective historically describes empirical evidence (so science) and deductive is used to describe logically coherent axiomic reasoning (which is obviously tautological as it’s a definition).
Greg you have walked yourself into the kind of linguistical mindfield that annoys you so much.
P.s I’m sure philosophy of mathematics has literature on precisely the argument you are making so it might be worth considering that before telling people that they don’t understand maths. It seems more likely that the Dunning Kruger is rearing its head but that is an inductive argument (or is it subjective).
If I am suffering from Dunning-Kruger then you should be able to refute my argument. If there are papers that demonstrate my ignorance then you should be able to point to them. For what it’s worth, any use by me of the term ‘objective’ is in the perfectly ordinary sense of ‘not influenced by personal feelings or opinions’ i.e. the opposite of subjective. This is not a preposterous way of describing deductive reasoning.
See here for some fun discourse on this topic: https://plato.stanford.edu/entries/philosophy-mathematics/
“Whereas the natural sciences investigate entities that are located in space and time, it is not at all obvious that this also the case of the objects that are studied in mathematics. In addition to that, the methods of investigation of mathematics differ markedly from the methods of investigation in the natural sciences. Whereas the latter acquire general knowledge using inductive methods, mathematical knowledge appears to be acquired in a different way: by deduction from basic principles. The status of mathematical knowledge also appears to differ from the status of knowledge in the natural sciences. The theories of the natural sciences appear to be less certain and more open to revision than mathematical theories.”
Quite.
Can I offer an example for the debate.
You can show that Pythagoras theorem is approximately true for lots of triangles by measuring them. Some teachers will do this.
You can also show that given a few axioms about plane geometry it is exactly true for all right angled triangles.
In the first case you have an inductive experiment and how far you generalize it is subjective – do you care about slight inaccuracies, do you care about very extreme cases – big, small, far from 45-90-45.
Is the first case math? Perhaps if it is a step to hinting that there is something to prove deductively using the second approach. But if you never take the second step you have some inductively derived result akin to a physics experiment. What would math be if this is how it was all done?
For over 2500 years people have had proofs of the theorem. If we don’t make that clear and distinguish it from the inductive approach something important is missed.
It may be that not all theorems will be proven before the existence and use of the theorem is taught but that these are things that are proven from axioms not derived experimentally can still be taught and understood.
just to be clear – my last statement referred proven in class.
I think this is a good illustration of the difference.
So to be clear it’s my job to prove you wrong despite the bold claim that you have made. This is not your normal position on evidence. You have swapped from objective/subjective to deductive/inductive which is dodgy. All this from countering Meyer who can not be simply wrong, instead his idea must be without foundation. A simple search shows plenty of discussion about non-deductive mathematical reasoning none of which would appease you as you have shifted the burden of proof and it would require emersion in a new field . It is obviously true that maths prefers deductive reasoning but the argument that it has no inductive elements is an ongoing debate (though incredibly fringe one) Have I misunderstood your argument as deductive equals objective? If I have I apologise if not then I reject it as chicanery. Your constantly moving the goalposts despite there being no need to do so. Meyer can have a point and still be wrong. It is a trivial thing to create a deductive argument which is objectively wrong. You simply use flawed premises. This is why formal deductive reasoning never took over the world. People reject the other side’s premises and even in mathematics contraversial axioms are still debated.
I haven’t given you a job. Do as you wish. If you want to disagree with me then this guidance may help (but feel free to ignore it):
http://www.paulgraham.com/disagree.html
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“…truths are only true if the axioms they are based upon are true and mathematics cannot be used to prove its own starting axioms.” – I will go down one step to define the truth first. The following three statements can be used to define the truths. (1) The laws of nature are the only truths. (2) The objects of nature and their characteristics create these laws. (3) The nature always demonstrates its all truths.
Mathematics begins with real numbers. But real numbers are not objects of nature. Therefore real numbers must be false. Mathematics assumes real numbers are points on a straight line. But there is no straight line in nature, because all objects in the universe are continuously moving.
Thus two of the fundamental objects of mathematics, real numbers and straight line, are invalid for nature. Therefore mathematics must be false for nature and cannot be used to describe the laws of nature or objects of nature, or their characteristics.
To illustrate – Apple is $10, cannot be justified by any means. Although this is what we widely use, but clearly it is not true. Note that money is also a real number, and therefore must be false too. Note that you may say, land is $1, labor is $2, etc., so adding up apple will be $10, but the original problem remains same – why land is $1.
Thus mathematics may be very consistent, very logical, inside its domain, but it cannot be used to describe nature and it will always fail for nature. Same is true for money.